Báo cáo hóa học: " Research Article OFDM Link Performance Analysis under Various Receiver Impairments" docx

Hindawi Publishing Corporation
EURASIP Journal on Wireless Communications and Networking
Volume 2008, Article ID 145279, 11 pages
doi:10.1155/2008/145279
Research Article
OFDM Link Performance Analysis under Various
Receiver Impairments
Marco Krondorf and Gerhard Fettweis
Vodafone Chair Mobile Communications Systems, Technische Universit
¨
at Dresden, D-01062 Dresden, Germany
Correspondence should be addressed to Marco Krondorf,
Received 8 May 2007; Accepted 11 September 2007
Recommended by Hikmet Sari
We present a methodology for OFDM link capacity and bit error rate calculation that jointly captures the aggregate effects of var-
ious real life receiver imperfections such as: carrier frequency offset, channel estimation error, outdated channel state information
due to time selective channel properties and flat receiver I/Q imbalance. Since such an analytical analysis is still missing in liter-
ature, we intend to provide a numerical tool for realistic OFDM performance evaluation that takes into account mobile channel
characteristics as well as multiple receiver antenna branches. In our main contribution, we derived the probability density function
(PDF) of the received frequency domain signal with respect to the mentioned impairments and use this PDF to numerically cal-
culate both bit error rate and OFDM link capacity. Finally, we illustrate which of the mentioned impairments has the most severe
impact on OFDM system performance.
Copyright © 2008 M. Krondorf and G. Fettweis. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
1. INTRODUCTION
Orthogonal frequency division multiplexing (OFDM) is
a widely applied technique for wireless communications,
which enables simple one-tap equalization by cyclic prefix in-
sertion. Conversely, the sensitivity of OFDM systems to vari-
ous receiver impairments is higher than that of single-carrier

systems. Furthermore, for OFDM system designers, it is of-
ten desirable to have easy to use numerical tools to predict
the system performance under various receiver impairments.
Within this article the term perfor mance meansbothlinkca-
pacity and uncoded bit-error rate (BER). Mostly, link level
simulations are used to obtain reliable performance measures
of a given system configuration. Unfortunately, simulations
are highly time consumptive especially when the parame-
ter space of the system under investigation is large. There-
fore, the intention of this article is to introduce a stochas-
tic/analytical method to predict the performance metrics of
a given OFDM system configuration. To get realistic perfor-
mance results, our approach takes into account a variety of
receiver characteristics and impairments as well as mobile
channel properties such as
(i) residual carrier frequency offset (CFO) after synchro-
nization;
(ii) channel estimation errors;
(iii) outdated channel state information due to time selec-
tive mobile channel properties;
(iv) flat receiver I/Q imbalance in case of direct conversion
receivers;
(v) frequency selective mobile channel characteristics;
(vi) multiple receiver branches to realize diversity com-
bining methods such as maximum ratio combining
(MRC).
In present OFDM standards, such as IEEE 802.11a/g or
DVB-T, preamble (or pilots) are used to estimate and to com-
pensate the CFO and channel impulse response. Unfortu-
nately, after CFO estimation and compensation, the resid-

ual carrier frequency offset still destroys the orthogonality of
the received OFDM signals and corrupts channel estimates,
which worsen further the performance of OFDM systems
during the equalization process. In the literature, the effects
of carrier frequency offset on bit-error rate are mostly in-
vestigated under the assumption of perfect channel knowl-
edge. The papers [5, 6] consider the effects of carrier fre-
quency offset only (without channel estimation and equal-
ization imperfections) and give exact analytical expressions
in terms of SNR-loss and OFDM bit-error rate for the AWGN
2 EURASIP Journal on Wireless Communications and Networking
channel. The authors of [8] extend the work of [5]toward
frequency-selective fading channels and derive the corre-
spondent bit-error rate for OFDM systems in case of CFO
under the assumption of perfect channel knowledge.
Cheon and Hong [1] tried to analyze the joint effects of
CFO and channel estimation error on uncoded bit-error rate
for OFDM systems, but the used Gaussian channel estima-
tion error model does not hold in real OFDM systems, espe-
cially when carrier frequency offset is large (see Section 5).
Additionally, receiver I/Q imbalance has been identified
as one of the most serious concerns in the practical imple-
mentation of direct conversion receiver architectures (see,
e.g., [12]). Direct conversion receiver designs are known to
enable small and cheap OFDM terminals, highly suitable for
consumer electronics. The authors of [11] investigated the
effect of receiver I/Q imbalance on OFDM systems for fre-
quency selective fading channels under the assumption of
perfect channel knowledge and perfect receiver synchroniza-
tion. Additionally, in order to cope with this impairment, the

authors of [10] proposed a digital I/Q imbalance compensa-
tion method.
To our best knowledge, there is currently no literature
available that describes a calculation method for OFDM BER
and link capacity under the aggregate effect of all the men-
tioned impairments. Therefore, our intention is to describe
the quantitative relationship between OFDM parameters, re-
ceiver impairments, and performance metrics such as bit-
error rate and link capacity. Furthermore, we intend to pro-
vide a useful system engineering tool for the design and
dimensioning of OFDM system parameters, pilot symbols,
and receiver algorithms used for frequency synchronization,
channel estimation, and I/Q imbalance compensation.
The structure of this article is as follows. After some
general remarks on our proposed link capacity evaluation
method in Section 2, we introduce our OFDM system model
in section followed by a general probability density function
analysis in Section 4.InSection 5, it will be explained how
to model the correlation between channel estimates and re-
ceived/impaired signals to derive uncoded bit-error rates of
OFDM systems with carrier frequency offset and I/Q imbal-
ance in Rayleigh frequency and time selective fading chan-
nels. It should be noted that the terms bit-error rate and bit-
error probability are used with equal meaning. This is due
to the fact that the bit-error rate converges toward bit-error
probability with increasing observation time in a stationary
environment. Finally, we introduce our link capacity calcula-
tion method in Section 6 and conclude in Section 7.
2. THE APPROACH
We choose link capacity, measured in bit/channel use, as an

important performance metric for OFDM system designs.
This information theoretic metric allows system designers to
characterize the system behavior subject to real-life receiver
impairments independently from any kind of channel coding
and iterative detection methods. As explained in Section 6
and illustrated in Figure 1, the OFDM transceiver chain in-
cluding channel and receiver properties can be characterized
as effective channel between source and detector, often called
the modulation channel. The modulation channel is charac-
terized by its conditional PDF f
Z|X
(z|x) that describes the
statistical relationship between the discrete input symbols x
and the continuously distributed decision variable z. Using
any given complex M-QAM constellation alphabet X, the
link capacity can be expressed as mutual information be-
tween source and sink that only depends on the input statistic
of X and f
Z|X
(z|x). Since our performance analysis frame-
work intends to describe the mutual information (and hence
the link capacity under a given input statistic), we propose
the following work flow.
(1) We show how to derive f
Z|X
(z|x) under receiver im-
pairments, given channel properties and OFDM sys-
tem parameters.
(2) We use the derived f
Z|X

(z|x) for uncoded BER calcu-
lation to verify its correctness by comparing the BER
prediction results with those obtained from simula-
tion.
(3) We calculate the mutual information, that is, OFDM
link capacity, using the verified statistic f
Z|X
(z|x).
3. OFDM SYSTEM MODEL
We consider an OFDM system with N-point FFT. The data
is M-QAM modulated to different OFDM data subcarriers,
then transformed to a time domain signal by IFFT operation
and prepended by a cyclic prefix, which is chosen to be longer
than the maximal channel impulse response (CIR) length L.
The sampled discrete complex baseband signal for the lth
subcarrier after the receiver FFT processing can be written
as
Y
l
= X
l
H
l
+ W
l
,(1)
where X
l
represents the transmitted complex QAM mod-
ulated symbol on subcarrier l,andW

l
represents complex
Gaussian noise. The coefficient H
l
denotes the frequency do-
main channel transfer function on subcarrier l, which is the
discrete Fourier transform (DFT) of the CIR h(τ)withmax-
imal L taps
H
l
=
L−1

τ=0
h(τ)e
−j2πlτ/N
. (2)
In this paper, it is assumed that the residual carrier frequency
offset (after frequency synchronization) is a given determin-
istic value. Furthermore, static (non-time-selective) channel
characteristics are assumed during one OFDM symbol. The
CFO-impaired complex baseband signal subcarrier l can be
written as
Y
l
= X
l
H
l
I(0) +

N/2−1

k=−N/2, k=l
X
k
H
k
I(k −l)+W
l
. (3)
The complex coefficients I(K
− l) represent the impact of
the received signal at subcarrier k on the received signal at
M. Krondorf and G. Fettweis 3
Coding &
symbol mapping
Discrete complex
input alphabet
X
OFDM
modulation
OFDM
demodulation &
combining
Mobile channel
SISO, SIMO
Continuous
complex detector
input alphabet
Z

Detection &
decoding
Modulation channel-representing PHY impairments,
modulation characteristics, and mobile channel properties
Figure 1: The modulation channel concept used for capacity evaluation.
subcarrier l due to the residual carrier frequency offsetasde-
fined in [5]
I(k
−l) = e
jπ((k−l)+Δ f )(1−1/N)
sin

π

(k − l)+Δ f

Nsin

π

(k − l)+Δ f

/N

,
(4)
where Δ f is the residual carrier frequency offset normalized
to the subcarrier spacing. In addition, later in this paper, the
summation


N/2−1
k=−N/2, k=l
will be abbreviated as

k=l
.In(3)
we can see that residual CFO causes a phase rotation of the
receivedsignal (I(0)) and intercarrier interference (ICI). Fur-
thermore, there is a time variant common phase shift for
allsubcarriersduetoCFOasgivenin[8] that is not mod-
eled here. This is due to the fact that this time variant com-
mon phase term is considered to be robustly estimated and
compensated by continuous pilots that are inserted among
the OFDM data symbols. I/Q imbalance of direct conversion
OFDM receivers directly translates to a mutual interference
between each pair of subcarriers located symmetrically with
respect to the DC carrier [10]. Hence, the received signal Y
l
at subcarrier l is interfered by the received signal Y
−l
at sub-
carrier
−l, and vice versa. Therefore, the undesirable leakage
due to I/Q imbalance can be modeled by [10, 12]

Y
l
= Y
l
+ K

l
Y
∗
−
l
,(5)
where (
·)
∗
represents the complex conjugation and K
l
de-
notes a complex-valued weighting factor that is determined
by the receiver phase and gain imbalance [10]. The image re-
jection capabilities of the receiver on subcarrier l can be ex-
pressed in terms of image rejection ratio (IRR) given by
IRR
l
=




1
K
l





2
. (6)
In this paper, we consider flat I/Q imbalance which sim-
ply means IRR
l
= IRR for all l. Subsequently, we con-
sider preamble-based frequency domain least-square (FDLS)
channel estimation to obtain the channel state information
(

H
l
) on subcarrier l:

H
l
=

Y
P,l
X
P,l
= I(0)H
l
+

k=l
X
P,k
H

k
I(k −l)+W

l
X
P,l
+ K
l

m
X
∗
P,m
H
∗
m
I
∗
(m + l)+W

−
l
X
P,l
,
(7)
where X
P,l
and


Y
P,l
denote the transmitted and received
preamble symbol on subcarrier l. The Gaussian noise of the
preamble part W

l
has the same variance as W
l
of the data
part (σ
2
W

l
= σ
2
W
l
). The channel estimate is used for frequency
domain zero-forcing equalization before data detection
Z
l
=

Y
l

H
l

,(8)
where Z
l
is the decision variable that is feed into the detec-
tor/decoder stage. The power of preamble signals and the av-
erage power of transmitted data signals on all carriers are
equivalent (
|X
P
|
2
= σ
2
X
). In case of multiple (N
Rx
)receiver
branches, maximum ratio combining (MRC) is used at the
receiver side. Therefore, the decision variable Z
l
on subcar-
rier l is given by
Z
l
=

N
Rx
κ=1
Y

l,k

H
∗
l,κ

N
Rx
κ=1



H
l,κ


2
,(9)
where κ denotes the receiver branch index. We assume that
there is the same IRR and CFO on all branches, what is
reasonable when considering one oscillator used for down-
conversion in each branch. Furthermore, we assume uncor-
related channel coefficients among the branches,
1
that is,
E

H
l,κ
1

H
∗
l,κ
2

= 0ifκ
1
=κ
2
, ∀l. (10)
3.1. Mobile channel characteristics
To obtain precise performance analysis results in case of sub-
carrier crosstalk induced by CFO and I/Q imbalance, it is
desirable to use exact expressions of the subcarrier channel
cross-correlation properties what is shown in more detail
in Section 5. The cross-correlation properties between fre-
quency domain channel coefficients are mainly determined
by the power delay profile of the channel impulse response
(CIR) and the CIR tap cross-correlation properties. Further-
more, the discrete nature of the sampled CIR is modeled
as tapped delay line having L channel taps. Although our
1
For sake of readability, we only include the antenna branch index κ if nec-
essary.
4 EURASIP Journal on Wireless Communications and Networking
analysis is not limited to a specific type of frequency selec-
tive channel, in our numerical examples, we consider mobile
channels having an exponential power delay profile (PDP):
σ
2

τ
=
1
C
e
−Dτ/L
, τ = 0, 1, , L − 1, (11)
where σ
2
τ
= E{|h(τ)|
2
} and the factor C =

L−1
τ=0
e
−Dτ/L
is
chosen to normalize the PDP as

L−1
τ
=0
σ
2
τ
= 1, what leads to
σ
2

H
= E{|H
l
|
2
}=1, for all l. The channel taps h(τ)areas-
sumed to be complex zero-mean Gaussian RV with uncorre-
lated real and imaginary parts. Hence, after DFT according
to (2), the channel coefficients are zero-mean complex Gaus-
sian random variables as well. Additionally, the CIR length L
is assumed to be shorter than/equal to the cyclic prefix. The
cross-correlation coefficient of the channel transfer function
on subcarriers k and l in case of frequency selective fading is
defined as
r
k,l
=
E

H
k
H
∗
l

σ
2
H
=1, ∀k=l, (12)
where σ

2
H
is equivalent for all subcarriers. Assuming mutual
uncorrelated channel taps of the CIR and applying (2), one
gets
E

H
k
H
∗
l

=
L−1

τ=0
σ
2
τ
e
−j2π(k−l)τ/N
. (13)
The cross-correlation property of the complex Gaussian
channel coefficients can be formulated to be
H
k
= r
k,l
H

l
+ V
k,l
, (14)
where V
k
is a complex zero-mean Gaussian with variance
σ
2
V
k,l
= σ
2
H
(1 −|r
k,l
|
2
)andE{V
k,l
H
∗
l
} = 0.
In current OFDM systems such as 802.11a/n or 802.16,
there is a typical OFDM block structure. An OFDM block
consists of a set of preamble symbols used for acquisition,
synchronization, and channel estimation, followed by a set
of serially concatenated OFDM data symbols. User mobil-
ity gives rise to a considerable variation of the mobile chan-

nel during one OFDM block (fast fading) what causes out-
dated channel information in certain OFDM symbols if there
is no appropriate channel tracking. To be precise, during the
time period λ between channel estimation and OFDM sym-
bol reception, the channel changes in a way that the esti-
mated channel information used for equalization does not
fit the actual channel anymore. If there is no channel track-
ing at the receiver side, our aim is to incorporate the effect of
outdated channel information into the performance analysis
framework. Therefore, we have to define the autocorrelation
properties of channel coefficients H
l
. The autocorrelation co-
efficient of subcarrier l is defined as follows:
r
H
(l,λ) =
E

H
l
(t)H
∗
l
(t + λ)

σ
2
H
. (15)

Applying (2)weget
E

H
l
(t)H
∗
l
(t + λ)

=
E

L−1

τ=0
L
−1

ν=0
h(τ, t)h
∗
(ν, t + λ)e
−2πl((τ−ν)/N)

.
(16)
When assuming uncorrelated channel taps, it follows
E


H
l
(t)H
∗
l
(t + λ)

=
L−1

τ=0
r
h
(τ, λ)σ
2
τ
. (17)
For sake of simplicity, it is assumed that all channel taps
have the same autocorrelation coefficient, that is, r
h
(τ, λ) =
r
h
(λ), for all 0 ≤ τ ≤ L − 1. Substituting the relation

L−1
τ
=0
σ
2

τ
= σ
2
H
and (16) into (15), we obtain
r
H
(l,λ) = r
h
(λ). (18)
For the numerical BER and link capacity evaluations done in
Section 5.2 and 6.2, the time selectivity of the complex Gaus-
sian channel taps was modeled as follows:
h(τ, t + λ)
= r
h
(τ, λ)h(τ, t)+v
τ,λ
, (19)
with
E



h(τ, t)


2

=

E



h(τ, t + λ)


2

=
σ
2
τ
, (20)
where v
τ,λ
is a complex Gaussian RV with variance σ
2
v
τ
,λ
=
σ
2
τ
(1 −|r
h
(τ, λ)|
2
)andE{h(τ, t)v

∗
τ,λ
}=0. For sake of sim-
plicity, it is assumed that the channel is stationary during
one OFDM symbol but changes from symbol to symbol in
the above defined manner. In our analysis, we intentionally
avoid any assumptions on concrete fast-fading models in or-
der to obtain fundamental results. Anyway, one of the com-
monly used statistical descriptions of fast channel variations
is the Jakes’ model [7], where the channel autocorrelation co-
efficient r
h
(τ)isgivenby
r
h
(τ) = J
0

2πf
D,max

, (21)
and f
D,max
denotes the maximum Doppler frequency that is
determined by the mobile velocity and carrier frequency of
the system. It should be noted that r
h
(τ) is real due to uncor-
related i.i.d. real and imaginary parts of the CIR taps.

4. PROBABILITY DENSITY FUNCTION ANALYSIS
The author of [9] suggested a correlation model regarding
channel estimation for single-carrier systems and derived the
correspondent symbol error-rate and bit-error rate of QAM-
modulated signals transmitted in flat Rayleigh and Ricean
channels. In this section, a short review of the contribution
of [9] will be given in order to further extend these results to
OFDM systems for time and frequency selective fading chan-
nels with CFO, I/Q imbalance, and channel estimation error.
The single-carrier transmission model without carrier fre-
quency offset for flat Rayleigh fading channels can be written
as
y
= hx + w, (22)
where y, h, x,andw denote the complex baseband repre-
sentation of the received signal, the channel coefficient, the
transmitted data symbol, and the additive Gaussian noise
M. Krondorf and G. Fettweis 5
with variance σ
2
w
,respectively.In[9], the channel estimate

h is assumed to be biased and used for zero forcing equaliza-
tion as follows:
z
=
y

h

with

h = αh + ν, (23)
where α denotes the deterministic multiplicative bias of
the channel estimates and ν represents zero-mean complex
Gaussian noise with variance σ
2
ν
. The channel coefficient h
and Gaussian noise ν are assumed to be uncorrelated. Hence,
the case of perfect channel knowledge can be easily modeled
by α
= 1andσ
2
ν
= 0.
In [9], the joint PDF of the decision variable z
= z
r
+ jz
i
in
case of transmit symbol x is derived in cartesian coordinates
and can be written as
f
Z|X
(z|x) =
a
2
(x)

π



z −b(x)


2
+ a
2
(x)

2




z=z
r
+jz
i
.
(24)
The PDF mainly depends on the complex parameter b(x),
given by [4, 9]
b(x)
= R{b} + jI{b}=b
r
(x)+jb
i

(x)
= x

α
∗
r
h
(λ)σ
2
h
|α|
2
σ
2
h
+ σ
2
ν

(25)
and the real parameter a(x) that can be written according
[4, 9]as
a
2
(x) =|x|
2
|α|
2
σ
4

h

1 − r
2
h
(λ)

+ σ
2
ν
σ
2
h

|
α|
2
σ
2
h
+ σ
2
ν

2
+
σ
2
w
|α|

2
σ
2
h
+ σ
2
ν
.
(26)
Additionally, the closed form integral of (24)withz
= z
r
+ jz
i
is given by [9]tobe
F
Z|X
(z|x)
=

z
i
−b
i
(x)

arctan


z

r
−b
r
(x)



a
2
(x)+

z
i
−b
i
(x)

2

2π

a
2
(x)+

z
i
−b
i
(x)


2
+

z
r
−b
r
(x)

arctan


z
i
−b
i
(x)



a
2
(x)+

z
r
−b
r
(x)


2

2π

a
2
(x)+

z
r
−b
r
(x)

2
.
(27)
In case of N
Rx
receiver branches, maximum ratio combining
(MRC) is used for decision variable computation what can
be formulated as
z
=

N
Rx
κ=1
y

κ

h
∗
κ

N
Rx
κ=1



h
κ


2
, (28)
where κ represents the antenna branch index, and the κth
channel estimate can be written according to the SISO case
as

h
κ
= α
k
h
κ
+ ν
κ

. (29)
x
2
(1, 0)
x
3
(1, 1)
B
1,1
x
4
(1, 0)

(
z)
x
1
(0, 0)

(
z)
Figure 2: The QPSK constellation digram, showing the decision re-
gion for one bit position of symbol x
1
.
Since it is quite reasonable to assume that the same channel
estimation scheme is used in each receive antenna branch, we
have α
κ
= α,forallκ.

The authors of [9] also derived the PDF of z in case of
transmit symbol x and N
Rx
receiver branches that is given by
f
Z|X,N
Rx

z|x, N
Rx

=
N
Rx

a
2
(x)

N
Rx
π



z −b(x)


2
+ a

2
(x)

N
Rx
+1
. (30)
It is easy to observe that the PDF (30) for the MRC case takes
the SISO form of (24)incaseofN
Rx
= 1. Additionally, the
closed form integral F
Z|X,N
Rx
(z|x, N
Rx
)off
Z|X,N
Rx
(z|x, N
Rx
)
can be found in [9] that also takes the SISO form (27)incase
of N
Rx
= 1. To enhance readability and to simplify our nota-
tion, we omit the receiver branch number N
Rx
in the condi-
tional PDF and its closed form integral, that is, in the follow-

ing we write f
Z|X
(z|x) instead of f
Z|X,N
Rx
(z|x, N
Rx
).
Finally, the result of (27) can be used to calculate the bit-
error rate of a given M-QAM constellation. In an M-QAM
constellation there are Mlog
2
(M)different possible bit posi-
tions with respect to the M-QAM constellation. The proba-
bility of an erroneous bit with respect to the mth QAM trans-
mit symbol x
m
can be calculated by using the closed form in-
tegral (27) and an appropriate decision region B
m,ν
for the
νth bit position (see Figure 2) that takes into account the bit
mapping of the QAM constellation. In the paper, we always
use Gray mapping in our numerical results, but it is worth
mentioning that the described method can be used for arbi-
trary bit mappings as well.
As already stated, we propose to use bit-error rate pre-
diction to verify the correctness of the derived probability
density function f
Z|X

(z|x) that is later used to determine
the OFDM link capacity of a given transceiver configuration.
Therefore, the bit-error probability P
b
(x
m
) takes the form
P
b

x
m

=
1
log
2
(M)
=
log
2
(M)

ν=1

F
Z|X

z|x
m


B
m,ν
, (31)
where [[F
Z|X
(z|x
m
)]]
B
m,ν
denotes the 2-dimensional evalua-
tion of the closed form integral F
Z|X
(z|x
m
) subject to the
6 EURASIP Journal on Wireless Communications and Networking
decision region B
m,ν
. Finally, the bit-error probability can be
obtained by averaging over all possible constellation points,
when assuming equal probable M-QAM symbols as follows:
P
b
=
1
M
M


m=1
P
b

x
m

. (32)
5. OFDM BIT-ERROR RATE ANALYSIS
In this section, the derivation of the bit-error rate of OFDM
systems with carrier frequency offset, I/Q imbalance, and
channel estimation error in Rayleigh frequency and time-
selective fading channels will be given. The central idea of
our BER derivation is to map the OFDM system model of
Section 3 to the statistics given in Section 4.Tobeprecise,we
have to map the OFDM system model to the parameters α,
a
2
(26)andb
2
(25) as explained below.
5.1. Mathematical derivation
Firstly, we can rewrite the channel estimates of subcarrier l in
(7) with respect to the frequency selective fading characteris-
tic given in (14)tobe

H
l
= I(0)H
l


1+

k=l
r
k,l
X
P,k
I(k −l)
I(0)X
P,l
+ e
K
l

m
r
∗
m,l
X
∗
P,m
I
∗
(m + l)
I(0)X
P,l

+ ν
l

,
(33)
where e denotes the term e
−j2φ
l
. This comes due to the fact
that the complex Gaussian channel coefficient can be written
as H
l
=|H
l
|e
jφ
l
.Hence,wehaveH
∗
l
/H
l
= e
−j2φ
l
= e,where
φ
l
is an equally distributed RV in the interval [−π : π]. From
(33)weobtainan(23)-like expression as follows:

H
l

= α
l

H
l
+ ν
l
, (34)
by defining effective channel

H
l
= I(0)H
l
and effective bias α
l
as
α
l
= 1+

k=l
r
k,l
X
P,k
I(k −l)+eK
l

m

r
∗
m,l
X
∗
P,m
I
∗
(m + l)
I(0)X
P,l
,
(35)
where
α
l
is a stochastic quantity with given subcarrier index
l, a set of deterministic preamble symbols X
P,k
,afixedpre-
determined frequency offset, a given IRR constant K
l
and RV
e
= e
j2φ
l
. It should be noted that the stochastic part of α
l
is

negligible in case of moderate I/Q imbalance (IRR
≥ 30 dB)
and moderate CFO. Hence, we have that
eK
l

m
r
∗
m,l
X
∗
P,m
I
∗
(m + l) ≈ 0, (36)
and α
l
can be well modeled to be a deterministic quantity.
This is due to the fact that the pilot symbols X
P,k
as well as the
CFO are given deterministic values and the channel cross-
correlation coefficients r
k,l
can be calculated using (12)and
(13).
The noise part
ν
l

of the channel estimate can be written
as
ν
l
= W

l
+ K
l
W

−
l
+

k=l
X
P,k
V
k,l
I(k −l)
X
P,l
+ K
l

m
X
∗
P,m

V
∗
m,l
I
∗
(m + l)
X
P,l
.
(37)
For σ
2
ν
l
, which represents the additive Gaussian noise vari-
ance of the channel estimates, we obtain
σ
2
ν
l
=

k=l

n=l
X
P,k
X
∗
P,n

I(k −l)I
∗
(n − l)

r
k,n
−r
k,l
r
∗
n,l
σ
2
H

+


K
l


2

k

n
X
∗
P,k

X
P,n
I
∗
(k + l)I(n + l)
×

r
∗
k,n
−r
∗
k,l
r
n,l
σ
2
H

+ σ
2
W

1+


K
l



2

.
(38)
Applying the same method as above for (3)and(5), the same
definition of effective channel

H
l
can be used to get a (22)-
like expression as follows:
Y
l
=

H
l

X
l
+

k=l
r
k,l
X
k
I(k−l)
I(0)
+ e

K
l

m
r
∗
m,l
X
∗
m
I
∗
(m + l)
I(0)

+

W
l
=

H

X
l
+

W
l
.

(39)
Given (39), the effective symbol

X
l
can be defined that is no
longer a deterministic value but a stochastic quantity due to
i.i.d. data symbols on subcarriers k
=l:

X
l
= X
l
+

k=l
r
k,l
X
k
I(k −l)+eK
l

m
r
∗
m,l
X
∗

m
I
∗
(m + l)
I(0)
  
stochastic part of the effective transmit symbol
.
(40)
Assuming a certain transmit symbol X
l
and assuming ran-
domly transmitted data symbols X
k
with k=l,wecandecom-
pose the effective symbol

X
l
as follows:

X
l
= X
l
+ J
l
, (41)
which shows the stochastic nature of


X
l
due to the random
interference part J
l
due to ICI and I/Q imbalance. Applying
the central limit theorem, we assume that the interference
J
l
term is a complex zero-mean Gaussian random variable
J
l
= p+ jq. The mutual uncorrelated real and imaginary parts
p and q have the same variance for all constellation points
σ
2
J
l
=

k=l


I(k −l)


2


r

k,l


2
+


K
l


2

m


I(m + l)


2


r
m,l


2
2



I(0)


2
.
(42)
M. Krondorf and G. Fettweis 7
According to (25)and(26), we calculate the parameters b
l
=
b
l,r
+ jb
l,i
and a
2
l
for M-QAM effective data symbols

X
l
on
subcarrier l in frequency and time selective fading channels:
b
l


X
l


=

X
l


α
∗
l
r
h
(λ)σ
2

H



α
l


2
σ
2

H
+ σ
2
ν


,
a
2
l


X
l

=



X
l


2



α
l


2
σ
4


H
l

1 − r
2
h
(λ)

+ σ
2
ν
l
σ
2

H
l

σ
2

H
l

2
+
σ
2

W

l
σ
2

H
l
,
(43)
where σ
2

H
l
=|I(0)|
2
σ
2
H
and σ
2

H
l
=|α
l
|
2
|I(0)|
2
σ

2
H
+ σ
2
ν
l
.From
(43) one can observe that the parameter σ
2

W
l
has to be cal-
culated exactly to obtain reliable results. The term

W
l
rep-
resents the effective noise of the received signal that consists
of AWGN parts W
l
, W
−l
, and ICI parts, respectively. If we
substitute (3)and(14) into (5), we get

W
l
= W
l

+ K
l
W
−l
+

k=l
X
k
V
k,l
I(k −l)
+K
l

m
X
∗
m
V
∗
m,l
I
∗
(m + l).
(44)
For an exact expression of σ
2

W

l
,wetake(44), σ
2
V
k,l
= σ
2
H
(1 −
|
r
k,l
|
2
) together with the assumptions of mutually uncorre-
lated data symbols and obtain
σ
2

W
l
= σ
2
W

1+


K
l



2

+ σ
2
H

k=l


I(k −l)


2

1 −


r
k,l


2

+


K
l



2
σ
2
H

m=l


I(m + l)


2

1 −


r
m,l


2

.
(45)
As an example, for one QPSK constellation point with index
m
= 1 on subcarrier l, X
1,l

= (1/
√
2)(1, 1) = (1/
√
2)(1+j), we
need to recalculate b
l
(

X
1,l
) and parameter a
2
l
(

X
1,l
) separately
for each effective symbol realization

X
1,l
= X
1,l
+ p + jq =
1
√
2
(1 + j)+p + jq (46)

to use the closed form integral and (31) for BER calculation.
Subsequently, the bit-error rate on subcarrier l for the mth
constellation point can be expressed using (31) by the fol-
lowing double integral involving the Gaussian PDFs of p and
q:
P
b

X
m,l

=

∞
−∞
P
b

X
m,l
+ p + jq

2πσ
2
J
l
e
−(p
2
+q

2
)/2σ
2
J
l
dpdq.
(47)
Finally, to obtain the general bit-error rate, we have to av-
erage (47)overallN
C
data subcarriers with index l and M-
QAM constellation points with index m as follows:

P
b
=
1
MN
C
N
C
/2−1

l=−N
C
/2
M

m=1
P

b

X
m,l

.
(48)
4035302520151050
SNR (dB)
Simulation
Δ f
= 1%, calculation
Δ f
= 5%, calculation
Δ f
= 7%, calculation
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
BER
MRC N

Rx
= 2
SISO
Figure 3: The comparison of simulated and calculated uncoded
BER versus SNR for 16-QAM OFDM under residual CFO in non-
time-selective channel environment and IRR
= 30 dB.
5.2. Bit-Error rate performance: numerical results
In this section, the derived analytical expressions for bit-error
rate are compared with appropriate simulation results for
both SISO (single-input single-output) OFDM transmission
as well as SIMO (single-input multiple-output) OFDM us-
ing MRC and two receiver antenna branches. Furthermore,
we consider an IEEE 802.11a-like OFDM system [3] with 64-
point FFT. The data is 16-QAM modulated to the data sub-
carriers, then transformed to the time domain by IFFT op-
eration and finally prepended by a 16-tap long cyclic prefix.
The data is randomly generated and one OFDM pilot symbol
was used for channel estimation. The used BPSK pilot data in
the frequency domain is given by
X
P,l
= (−1)
l
for subcarrier index l = [−26 : 1 : 26], l=0.
(49)
The data and pilot symbols are modulated on 52 data carri-
ers. The DC carrier as well as the carriers at the spectral edges
are not modulated and are often called “virtual carriers.” F o r
simulation and numerical BER analysis, we use an 8 taps ex-

ponential PDP frequency selective Rayleigh fading channel
with D
= 7 (see Section 3). Furthermore, we choose statis-
tical independent channel realizations for the two antenna
branches in case of SIMO OFDM transmission.
Thedoubleintegralof(47)isevaluatednumericallyus-
ing Matlab built-in integration functions having a numeri-
cal tolerance of 10
−8
and upper/lower integration bounds of
±10.
Figure 3 illustrates the calculated and simulated 16-QAM
BER versus SNR (σ
2
X
/σ
2
W
) with given carrier frequency offset
Δ f (in % subcarrier spacing) and IRR
= 30 dB under non-
time variant mobile channel conditions.
Figure 4 illustrates the calculated and simulated 16-QAM
BER versus SNR (σ
2
X
/σ
2
W
) with given carrierfrequency offset

8 EURASIP Journal on Wireless Communications and Networking
4035302520151050
SNR (dB)
IRR
= 30 dB, simulation
IRR
= 40 dB, simulation
IRR
= 30 dB, calculation
IRR
= 40 dB, calculation
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
BER
MRC N
Rx
= 2
SISO

Figure 4: The comparison of simulated and calculated uncoded
BER versus SNR for 16-QAM OFDM with residual CFO of 3% un-
der non-time selective channel conditions under IRR
= 30 dB/40 dB.
4035302520151050
SNR (dB)
Simulation
r
h
(λ) = 0.99, calculation
r
h
(λ) = 0.995, calculation
r
h
(λ) = 0.998, calculation
10
−4
10
−3
10
−2
10
−1
10
0
BER
MRC N
Rx
= 2

SISO
Figure 5: The comparison of simulated and calculated uncoded
BER versus SNR for 16-QAM OFDM with residual CFO of 3% and
IRR
= 30 dB under time selective channel conditions.
Δ f (in % subcarrier spacing) and IRR = 30 dB under non-
time variant mobile channel conditions.
In Figure 5, we use a fixed Δ f of 3% to investigate 16-
QAM BER versus SNR for time variant mobile channel prop-
erties, characterized by the channel tap autocorrelation coef-
ficients r
h
(λ).
The results illustrate that our analysis can approximate
the simulative performance very accurately if the channel
power delay profile, the image rejection ratio of the direct
conversion receiver, and carrier frequency offset are known.
6. CAPACITY ANALYSIS OF IMPAIRED OFDM LINKS
To perform OFDM link capacity analysis, it seems manda-
tory to review the main principles and basic equations of how
to calculate average mutual information between source and
sink of a modulation channel. An excellent overview of this
topic can be found in [7] that is summarized in the following.
In an OFDM system, we have a number of parallel channels,
that is, data subcarriers. Hence we propose to calculate the
mutual information for each of the parallel data carriers in-
dependently and to finally average the link capacity among
the data carriers.
Let us consider real input and output alphabets X and Z.
Both alphabets can be characterized in terms of information

content carried by the elements of each alphabet what leads
to the concept of information entropy H(X) and H(Z).The
entropy of the discrete alphabet X having elements X
m
with
appropriate probability P(X
m
)isgivenby
H(X) =−

m
P

X
m

log
2

P

X
m

. (50)
Conversely, Z is assumed to be a real continuously dis-
tributed RV having realizations z. As a result, Z can be char-
acterized by its differential entropy as
H(Z)
=−


Z
f
Z
(z)log
2

f
Z
(z)

dz, (51)
where f
Z
(z) denotes the PDF of Z. Finally, the mutual infor-
mation I(X; Z)ofX and Z can be formulated as [7]
I(X; Z)
=

m
P

X
m


Z
f
Z|X


z|X
m

×
log
2

f
Z|X

z|X
m


n
f
Z|X

z|X
n

P

X
n


dz.
(52)
Itcanbeseenfrom(52) that I(X; Z) requires knowledge of a-

priory probabilities P(X
m
) and conditional PDFs f
Z|X
(z|X
m
)
only. Mostly we have that P(X
m
) = 1/M in case of M-ary con-
stellations. Since the above defined mutual information cal-
culation scheme assumes one-dimensional output variables
and z is a two-dimensional complex RV of real part z
r
and
imaginary part z
i
,wehavetosolveadoubleintegraltoob-
tain the corresponding mutual information as follows:
I(X; Z)
=

m
P

X
m


Z

r

Z
i
f
Z|X

z
r
+ jz
i
| X
m

×
log
2

f
Z|X

z
r
+ jz
i
|X
m


n

f
Z|X

z
r
+ jz
i
| X
n

P

X
n


dz
r
d
Z
i
.
(53)
6.1. Mutual information under carrier crosstalk
Recalling the two-dimensional conditional SISO PDF
f
Z|X
(z
r
+ jz

i
| X
m
) on subcarrier l as given in Section 4,we
have that
f
Z|X

z
r
+ jz
i
| X
m

=
a
2
l

X
m

π



z
r
+ jz

i
−b
l

X
m



2
+ a
2
l

X
m

2
,
(54)
M. Krondorf and G. Fettweis 9
where a
2
l
(X)andb
l
(X) contain the entire OFDM link im-
pairment information (channel estimation error, I/Q im-
balance, CFO, outdated channel information, and channel
power delay profile). According to Section 4, the complex-

valued transmit symbol is stochastic by nature due to CFO
and I/Q imbalance carrier crosstalk and can be expressed as
X
m
+ J = X
m
+ p + jq,wherem represents the constellation
point index while p and q represent the effects of I/Q imbal-
ance and residual CFO. Both, p and q can be modeled as i.i.d.
zero-mean Gaussian RV as done in Section 4. Additionally,
both parameters a
2
l
(X)andb
l
(X) are subcarrier-dependent.
As a result (54) has to be reformulated for subcarrier l as
f
Z|X,P,Q

z
r
+ jz
i
| X
m
+ p + jq

=
a

2
l

X
m
+ p + jq

π



z
r
+ jz
i
−b
l

X
m
+ p + jq



2
+ a
2
l

X

m
+ p + jq

2
.
(55)
Hence, the calculation of p/q-independent conditional
marginal PDFs can be done via numerical double integration
as
f
Z|X

z
r
+ jz
i
| X
m

=

∞
−∞
f
Z|X,P,Q

z
r
+ jz
i

| X
m
+ p + jq

×
f
Q
(q) f
P
(p)dpdq.
(56)
According to Section 5, we have the Gaussian distribution for
each p and q:
f
P
(p) = f
Q
(q) =
1

2πσ
2
J
e
−(p,q)
2
/2σ
2
J
, (57)

where σ
2
J
isgivenin(42). In case of MRC multiantenna re-
ception, we have to proceed in the same manner.
6.2. OFDM link capacity: numerical examples
The quantitative relationship between receiver impairments,
OFDM system parameters and link capacity is an essential
piece of information for the dimensioning of I/Q imbalance
compensation algorithms as well as frequency synchroniza-
tion methods. Moreover, the effects of time-selective mobile
channels on link capacity can be used to design scattered pi-
lot structures for channel estimation and tracking as done
in [2]. Generally, link capacity indicates the maximum data
rate that can be achieved with strong channel coding under
a given input constellation and a specified receiver architec-
ture.
The numerical examples of average mutual information
are chosen such that we illustrate the effects of channel es-
timation error, outdated channel state information (CSI),
residual CFO, and flat receiver I/Q imbalance on the link ca-
pacity of SISO and SIMO OFDM links. Therefore, we choose
the same IEEE 802.11a-like OFDM system parameters as in-
troduced in Section 5.2, assume an 8 taps exponential PDP
mobile channel and the use of 16-QAM modulation on each
data carrier. Again, statistical independent channel realiza-
tions for the N
RX
antenna branches in case of SIMO OFDM
302520151050−5−10

SNR (dB)
SISO, perfect CSI
MRC N
Rx
= 2, perfect CSI
SISO, FDLS
MRC N
Rx
= 2, FDLS
0
0.5
1
1.5
2
2.5
3
3.5
4
Mutual information in bit/channel use
16-QAM upper bound
Figure 6: The mutual information, averaged over all data carri-
ers, comparison between perfect channel-state information and real
FDLS channel estimation for SISO and SIMO OFDM, CFO
= 0%,
no I/Q imbalance, static Rayleigh fading channel.
transmission are assumed. The mutual information (mea-
sured in Bit/Channel Use) is averaged among the data car-
riers and plotted over SNR (σ
2
X

/σ
2
W
).
In Figure 6, we illustrate the effect of real-life frequency
domain least-square (FDLS) channel estimation on the link
capacity of SISO and SIMO OFDM, respectively, assuming
no I/Q imbalance, a perfect frequency synchronization (CFO
= 0%) and static (non-time-selective) channel properties. As
reference, we plotted the case of perfect channel state infor-
mation that can easily be modelled by α
l
= 1andσ
2
ν
l
= 0.
In Figure 7, we show the aggregate effect of I/Q imbal-
ance and FDLS channel estimation under static-channel con-
ditions and perfect frequency synchronization. It is easy to
see that I/Q imbalance has only little effect on the averaged
mutual information performance, what is especially the case
at realistic image rejection ratios above 30 dB. Interestingly,
a worst case IRR of 20 dB heavily impacts the SISO perfor-
mance but causes only a small performance loss in case of
receiver diversity combining.
Figure 8 depicts the effect of CFO on averaged link capac-
ity under real FDLS channel estimation and no I/Q imbal-
ance under static-channel conditions. It can be shown that a
moderate CFO of 3% causes only a negligable degradation of

SISO and SIMO OFDM link capacity. The worst case perfor-
mance in case of CFO
= 10% is plotted to illustrate the lower
sensitivity of the SIMO link compared to the SISO link. Nev-
ertheless, we have to state that in case of realistic frequency
synchronization techniques, it is highly improbable to have a
residual CFO larger than 3% at moderate SNR (> 10dB).
This fact is also mentioned in [4] where the authors de-
rived the PDF of the residual CFO in case of real frequency
synchronization under Rayleigh fading channels and given
SNR.
10 EURASIP Journal on Wireless Communications and Networking
35302520151050
SNR (dB)
No I/Q imbalance
IRR
= 30 dB
IRR
= 20 dB
0.5
1
1.5
2
2.5
3
3.5
4
Mutual information in bit/channel use
16-QAM upper bound
MRC N

Rx
= 2
SISO
Figure 7: The mutual information averaged over all data carriers
under the aggregate effect of I/Q imbalance and FDLS channel es-
timation for SISO OFDM, 16-QAM, CFO
= 0%, static 8 taps expo-
nential PDP Rayleigh fading channel.
35302520151050
SNR (dB)
No CFO
CFO
= 3%
CFO
= 10%
0.5
1
1.5
2
2.5
3
3.5
4
Mutual information in bit/channel use
16-QAM upper bound
MRC N
Rx
= 2
SISO
Figure 8: The mutual information averaged over all data carriers

under CFO, FDLS channel estimation is assumed, 16-QAM modu-
lation on all subcarriers, no I/Q imbalance, time variant 8 taps ex-
ponential PDP Rayleigh fading channel.
Figure 9 depicts the effect of outdated channel-state in-
formation quantified by appropriate channel autocorrelation
coefficients r
h
(λ), FDLS channel estimation and I/Q imbal-
ance under 8 taps exponential PDP Rayleigh fading channel
conditions and perfect frequency synchronization. Again, the
performance loss in case of diversity combining is smaller
than the loss that we have in case of conventional SISO
receiver designs. Moreover, we have to state that even in case
3025201510
SNR (dB)
r
h
(λ) = 1
r
h
(λ) = 0.995
r
h
(λ) = 0.99
r
h
(λ) = 0.985
0.5
1
1.5

2
2.5
3
3.5
4
Mutual information in bit/channel use
16-QAM upper bound
MRC N
Rx
= 2
SISO
Figure 9: The mutual information averaged over all data carriers
under time-selective channel properties and FDLS channel estima-
tion for SISO OFDM, 16-QAM, CFO
= 0%, IRR = 30 dB, time vari-
ant 8 taps exponential PDP Rayleigh fading channel.
302520151050
SNR (dB)
CFO
= 3%, IRR = 30 dB and real FDLS
CFO
= 3%, IRR = 30 dB and perfect CSI
No impaiments, perfect CSI
0
0.5
1
1.5
2
2.5
3

3.5
4
Mutual information in bit/channel use
16-QAM upper bound
MRC N
Rx
= 2
SISO
Figure 10: The mutual information averaged over all data carriers,
comparing the effect of receiver impairments in case of perfect CSI
and real FDLS channel estimation, 16-QAM modulation on all sub-
carriers, static 8 taps exponential PDP Rayleigh fading channel.
of very small deviations of r
h
(λ) from the ideal static case
r
h
(λ) = 1, the effect of outdated channel-state information
causes much larger performance losses than realistic CFO
and I/Q imbalance.
Finally, we want to highlight the fact that in case of
moderate receiver impairments the performance loss mainly
comes due to channel-estimation errors. This important
observation is illustrated in Figure 10 where we plotted
M. Krondorf and G. Fettweis 11
averaged mutual information versus SNR under CFO = 3%
and IRR
= 30 dB assuming static channel properties. As ref-
erence we use a plot without any I/Q imbalance, CFO, or
channel estimation error. Interestingly the impairment plots

in case of perfect CSI are almost equivalent to the reference
curves but we observe a severe performance degradation in
case of real FDLS channel estimation.
7. CONCLUSIONS
In this paper, we show how to analytically evaluate the un-
coded bit-error rate as well as link capacity of OFDM sys-
tems subject to carrier frequency offset, channel estimation
error, outdated channel state information, and flat receiver
I/Q imbalance in Rayleigh frequency and time-selective mo-
bile fading channels. The probability density function of the
frequency domain received signal subject to the mentioned
impairments is derived. Furthermore, this PDF is verified by
means of bit-error rate calculation. We show that our ap-
proach can be used to exactly evaluate uncoded bit-error
rates when a priori knowledge of the mobile channel power
delay profile, the image rejection ratio and receiver CFO is
used. Furthermore, we show how to use the derived PDF to
calculate OFDM link capacity under the aggregate effects of
receiver impairments and mobile channel characteristics. Fi-
nally, we highlight the fact that channel uncertainty induced
by channel estimation errors as well as outdated channel state
information have much severer impact on OFDM capacity
than CFO or I/Q imbalance.
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